On the de-Rham cohomology of hyperelliptic curves
On the de-Rham cohomology of hyperelliptic curves
For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of \v{C}ech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.
hyperelliptic curve, de-Rham cohomology, hyperelliptic involution, Hodge-de-Rham short exact sequence, \v{C}ech cohmology
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Tait, Joseph
08f47697-9060-4958-b69a-facdafc478bc
2018
Koeck, Bernhard
84d11519-7828-43a6-852b-0c1b80edeef9
Tait, Joseph
08f47697-9060-4958-b69a-facdafc478bc
Koeck, Bernhard and Tait, Joseph
(2018)
On the de-Rham cohomology of hyperelliptic curves.
Research in Number Theory, 4 (2), [19].
(doi:10.1007/s40993-018-0111-4).
Abstract
For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of \v{C}ech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.
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Accepted/In Press date: 15 March 2018
e-pub ahead of print date: 2 April 2018
Published date: 2018
Keywords:
hyperelliptic curve, de-Rham cohomology, hyperelliptic involution, Hodge-de-Rham short exact sequence, \v{C}ech cohmology
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Local EPrints ID: 413972
URI: http://eprints.soton.ac.uk/id/eprint/413972
ISSN: 2363-9555
PURE UUID: 72da5c64-58f4-4b4f-bd47-44b664a2f076
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Date deposited: 12 Sep 2017 16:31
Last modified: 16 Mar 2024 03:22
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Author:
Joseph Tait
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